few_shots: Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001097

7
options: Summarize, represent, and interpret data on two categorical and quantitative variables.

Interpret linear models.

Understand and evaluate random processes underlying statistical experiments.

Understand independence and conditional probability and use them to interpret data.

Summarize, represent, and interpret data on a single count or measurement variable.

Investigate chance processes and develop, use, and evaluate probability models.

Use probability to evaluate outcomes of decisions.

Use random sampling to draw inferences about a population.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Develop understanding of statistical variability.

Investigate patterns of association in bivariate data.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Summarize and describe distributions.

Calculate expected values and use them to solve problems.

Draw informal comparative inferences about two populations.
problem_activity: Activity
The mathematical purpose of this activity is for students to understand that approximately normal distributions have a similar proportion of area in regions described by the mean and standard deviations. Students are given a data set including the number of words for 200 submitted short stories. They are asked to determine the proportion of submitted stories that are within certain ranges.
Monitor for students who:
return to the original data set for each question and count the values in each interval
create a frequency table to break the data into natural intervals
create a histogram to visualize the areas in question
use technology to count values in each region
The problem mentions that a page holds approximately 200 words to help students identify natural intervals to summarize the data into a frequency table or histogram.
Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Digital
Launch
Arrange students in groups of 2.
Ask, “What do you think the phrase ‘within two standard deviations of the mean’ means?” (It means the interval from the value of the mean minus two times the standard deviation to the value of the mean plus two times the standard deviation.)
If students are using a computer to create a spreadsheet, histogram, or find other information, suggest that they copy and paste the data into a program like Geogebra, Desmos, or other spreadsheet program rather than type all of the data themselves.
Student Facing
A publisher takes submissions for short stories to include in a book. 200 stories are submitted, but the publisher needs to be aware of how long each story is. The way the publisher will put together the collection of stories, a page typically contains 200 words. The mean number of words for each story is 2,600 and the standard deviation is 400 words.
2844
2643
3316
2084
2316
2513
2931
2563
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2465
2821
2493
2263
2706
2501
2627
2220
2372
2635
3066
2824
2357
2522
2564
2901
2118
2325
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2073
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1976
2746
3462
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1952
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3054
2476
2853
2538
2167
2732
3304
2347
3015
2151
2446
2714
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2727
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2367
3116
2650
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2946
1849
2897
2625
2938
2407
2218
2287
2356
2125
3296
2289
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2715
2793
2631
2973
2876
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3259
3094
2452
2149
3043
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2542
2753
2985
2501
2393
2896
2135
3191
2319
1984
2013
2462
3186
2674
2273
2483
2671
2702
2819
2197
2427
2018
1927
2428
2438
1852
2395
1826
2767
If a histogram is created using intervals of 200 words, what would be the area of the bar representing the number of stories that contain between 2,000 and 2,200 words? Explain or show your reasoning.
What proportion of the total area is represented by the bar for stories that contain between 2,000 and 2,200 words? Explain or show your reasoning.
What proportion of stories in this group contain between 2,000 and 2,200 words? Explain or show your reasoning.
How does the proportion of the area you calculated relate to the proportion of stories in the group that contain between 2,000 and 2,200 words?
What proportion of stories in this group are within 1 standard deviation of the mean number of words?
What proportion of stories in this group are within 2 standard deviations of the mean number of words?
What proportion of stories in this group are within 1 standard deviation of 2,400 words?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2. Distribute one copy of the blackline master to each group.
Ask, “What do you think the phrase ‘within two standard deviations of the mean’ means?” (It means the interval from the value of the mean minus two times the standard deviation to the value of the mean plus two times the standard deviation.)
Student Facing
A publisher takes submissions for short stories to include in a book. 200 stories are submitted, but the publisher needs to be aware of how long each story is. The way the publisher will put together the collection of stories, a page typically contains 200 words. The mean number of words for each story is 2,600 and the standard deviation is 400 words.
If a histogram is created using intervals of 200 words, what would be the area of the bar representing the number of stories that contain between 2,000 and 2,200 words? Explain or show your reasoning.
What proportion of the total area is represented by the bar for stories that contain between 2,000 and 2,200 words? Explain or show your reasoning.
What proportion of stories in this group contain between 2,000 and 2,200 words? Explain or show your reasoning.
How does the proportion of the area you calculated relate to the proportion of stories in the group that contain between 2,000 and 2,200 words?
What proportion of stories in this group are within 1 standard deviation of the mean number of words?
What proportion of stories in this group are within 2 standard deviations of the mean number of words?
What proportion of stories in this group are within 1 standard deviation of 2,400 words?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Prove more generally that the proportion of total area taken up by a bar in a histogram is equal to the proportion of all data values that are contained in the interval represented by bar. To begin, let
\(n\)
represent the number of data values in an interval given by one bar,
\(M\)
represent the number of data values in the entire set, and
\(w\)
be the width of the interval in each bar of the histogram. Prove that the proportion of area taken up by the bar is
\(\frac{n}{M}\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may have trouble understanding the phrase “within 1 standard deviation of the mean.” Ask students to draw a number line containing the mean value and mark on the number line 1 standard deviation below and 1 standard deviation above the mean.
Activity Synthesis
The goal of this discussion is to get students thinking about histograms as an area model for a distribution. Select previously identified students to share in this order:
return to the original data set for each question and count the values in each interval
create a frequency table to break the data into natural intervals
create a histogram to visualize the areas in question
use technology to sort or count values in each interval
Connect the methods used by asking students about the advantages and disadvantages of the methods they used. (The summary methods, such as a frequency table or histogram, require additional time to set up, but are quicker to find the useful values for questions such as the ones included in the activity. Using technology, such as spreadsheets, to sort or count the data is also quick and accurate, but may take some time to enter the data if it is not already on the computer.)
If possible, display a histogram created by the students. If none of the students created a histogram, display the one here.
[IMG]
Here are some questions for discussion.
“How is this activity related to the previous one?” (In the previous activity, we figured out the area of a shaded region and divided it by the total area under the curve. In this activity, we can think of the bars as having an area.)
“Why would a normal distribution be a good model for the data summarized in the histogram?” (The distribution is approximately symmetric and bell-shaped.)
“Why do you think the area for a bar representing 2,000 to 2,200 words has the same proportion of the whole as the proportion of stories in the same region to the whole?” (Since the area came from
\(\frac{18 \boldcdot 200}{200 \boldcdot 200}\)
, which is equivalent to
\(\frac{18}{200}\)
.)
“Was it surprising to you that more than 95% of the data was within two standard deviations of the mean or that almost 70% of the data was within one standard deviation of the mean? Explain your reasoning.” (Both were surprising to me. I had never really thought about using the standard deviation to create an interval around the mean, so I had never thought about using the standard deviation to estimate the proportion of the data that was within a specific interval. But since most of the data is near the center, I can understand why it is this way.)
“Why do you think it could be useful to look at the proportion of values that are within one standard deviation of the mean?” (In the last lesson, we were introduced to normal curves that were based on the mean and standard deviation. If we modeled a histogram with a standard curve, it would be interesting to know what proportion of values fall within a given interval on normal curves.)


correct_option_index: 4